Lattices of minimum covolume in Chevalley groups over local fields of positive characteristic

TL;DR

This paper proves that for certain Chevalley groups over local fields of positive characteristic, the lattice of minimum covolume is unique and explicitly identified as G(F_q[t]) under specific conditions.

Contribution

It establishes the uniqueness of the minimal covolume lattice in specific Chevalley groups over local fields of positive characteristic.

Findings

Unique minimal covolume lattice identified as G(F_q[t])

Conditions on q and group type for the result to hold

Extension of covolume classification to new classes of Chevalley groups

Abstract

In this article, we show that if G is a simply connected Chevalley group of either classical type of rank bigger than 1 or type E6, and q > 9 is a power of a prime number p > 5, then G = G(F_q((1/t))), up to an automorphism, has a unique lattice of minimum covolume, which is G(F_q[t]).